### Abstract

If x, y, z are real numbers satisfying x + y + z = 1, then the maximum of the quadratic form axy + bxz + cyz with positive constants a, b, c is abc/2ab + 2ac + 2bc − a^{2} − b^{2} − c^{2} under the assumption √a < √b + √c. Extending this fact, we give the maximum of the quadratic form ∑1≤i<j≤n a_{ij}x_{i}x_{j} in n-variables x_{1}, …, x_{n} satisfying ∑^{n} _{i=1} x_{i} = 1 with constants a_{ij} = 0 under certain assumptions.

Original language | English |
---|---|

Pages (from-to) | 641-658 |

Number of pages | 18 |

Journal | Hokkaido Mathematical Journal |

Volume | 35 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 2006 |

Externally published | Yes |

### Fingerprint

### Keywords

- Distance matrix
- Ozeki’s inequality
- Quadratic form

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Hokkaido Mathematical Journal*,

*35*(3), 641-658. https://doi.org/10.14492/hokmj/1285766422

**Maximization of quadratic forms expressed by distance matrices.** / Izumino, Saichi; Nakamura, Noboru.

Research output: Contribution to journal › Article

*Hokkaido Mathematical Journal*, vol. 35, no. 3, pp. 641-658. https://doi.org/10.14492/hokmj/1285766422

}

TY - JOUR

T1 - Maximization of quadratic forms expressed by distance matrices

AU - Izumino, Saichi

AU - Nakamura, Noboru

PY - 2006/1/1

Y1 - 2006/1/1

N2 - If x, y, z are real numbers satisfying x + y + z = 1, then the maximum of the quadratic form axy + bxz + cyz with positive constants a, b, c is abc/2ab + 2ac + 2bc − a2 − b2 − c2 under the assumption √a < √b + √c. Extending this fact, we give the maximum of the quadratic form ∑1≤iijxixj in n-variables x1, …, xn satisfying ∑n i=1 xi = 1 with constants aij = 0 under certain assumptions.

AB - If x, y, z are real numbers satisfying x + y + z = 1, then the maximum of the quadratic form axy + bxz + cyz with positive constants a, b, c is abc/2ab + 2ac + 2bc − a2 − b2 − c2 under the assumption √a < √b + √c. Extending this fact, we give the maximum of the quadratic form ∑1≤iijxixj in n-variables x1, …, xn satisfying ∑n i=1 xi = 1 with constants aij = 0 under certain assumptions.

KW - Distance matrix

KW - Ozeki’s inequality

KW - Quadratic form

UR - http://www.scopus.com/inward/record.url?scp=85035292234&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85035292234&partnerID=8YFLogxK

U2 - 10.14492/hokmj/1285766422

DO - 10.14492/hokmj/1285766422

M3 - Article

AN - SCOPUS:85035292234

VL - 35

SP - 641

EP - 658

JO - Hokkaido Mathematical Journal

JF - Hokkaido Mathematical Journal

SN - 0385-4035

IS - 3

ER -